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- Math1026 - Sets, Sequences And Series

Roberto F.

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65
Field

A set that satisfies the following axioms:

A1) Associative

A2) Commutative

A3) Additive and multiplicative identity

A4) Additive inverses

A5) Multiplicative inverses

A6) Distributivity

Ordered Field

A field which satisfies the following axioms:

A1) Trichotemy

A2) Addition Law

A3) Multiplicative Law

A4) Transitivity

Bounded Above

Let K be an ordered field. Then a subset of K (S) is called bounded above if there is an element M in K such that x in M such that x=<M for all x in S. This M is the upper bound

Bounded Below

A subset of K (S) is bounded below iff -S is bounded above

If the upper/lower bound equals infinite

The set is not bounded

Least Upper Bound

If there is no upper bound that is smaller than M

If a subset of K (S) is finite....

Complete (Ordered) Field

Supremum, Sup S

Infimum, Inf S

Archimedean Property of R

Density of Q in R

Sequence of Reals

Constant Sequence

Monotonously Increasing Sequences

Strictly Monotonously Increasing Function

Monotonously Decreasing Sequence

Monotonously Strictly Decreasing Sequence

Monotonous Sequence

Bounded

Bounded (Sequence)

Limit (Infinity)

Convergence

A convergent sequence has....

How to prove a sequence converges:

Any convergent sequence...

Limit Theorems

The Squeeze Theorem

Monotone Convergence Theorem

Reverse Monotone Convergence Theorem

Closed Sequence

Nested Sequence Lemma

Subsequence

If a_{n}->L and (b_{k}) is a subsequence of (a_{n})...

Bolzano-Weierstrass

Theorem

Theorem

Every sequence....

Cauchy Sequences

Every Cauchy is...

Let (a_{n}) be Cauchy, and assume some subsequence of (a_{n}) converges to L.

A Real Sequence Converges iff...

Series

Partial Sums

A series converges...

If a series does not converge, then...

Harmonic Series

Dominant Term

A series converges iff

The Divergence Test

Comparison test

Ratio Test

The Alternating Series Test

Let (a_{n}) be a sequence such that a_{2k}->L and a_{2k+1}->L, then...

Absolute Convergence

Absolute Convergence

Sequential Continuity

Discontinuous

Continuous

Algebra of Continuous Functions

Rational Functions

The Composition of Two Contionus Functions...

Intermediate Value Theorem

Strictly Increasing Function

x^{1/p}...

Extreme Value Theorem

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